Stochastic service network designs with uncertain demand represented by a set of scenarios can be modelled as a large-scale two-stage stochastic mixed-integer program (SMIP). The progressive hedging algorithm (PHA) is a decomposition method for solving the resulting SMIP. The computational performance of the PHA can be greatly enhanced by decomposing according to scenario bundles instead of individual scenarios. At the heart of bundle-based decomposition is the method for grouping the scenarios into bundles. In this paper, we present a fuzzy c-means-based scenario bundling method to address this problem. Rather than full membership of a bundle, which is typically the case in existing scenario bundling strategies such as k-means, a scenario has partial membership in each of the bundles and can be assigned to more than one bundle in our method.

Physical design and production of Integrated Circuits (IC) is becoming increasingly more challenging as the sophistication in IC technology is steadily increasing. Placement has been one of the most critical steps in IC physical design. Through decades of research, partition-based, analytical-based and annealing-based placers have been enriching the placement solution toolbox. However, open challenges including long run time and lack of ability to generalize continue to restrict wider applications of existing placement tools. We devise a learning-based placement tool based on cyclic application of Reinforcement Learning (RL) and Simulated Annealing (SA) by leveraging the advancement of RL. Results show that the RL module is able to provide a better initialization for SA and thus leads to a better final placement design. Compared to other recent learning-based placers, our method is majorly different with its combination of RL and SA. It leverages the RL model's ability to quickly get a good rough solution after training and the heuristic's ability to realize greedy improvements in the solution.

In the simplest version of this problem, given access to a graph, one seeks to cluster the vertices into interpretable communities or groups of vertices, which are believed to reflect latent similarities among the nodes. From a theoretical standpoint, this problem has been extensively analyzed under specific generative assumptions on the observed graph; the most popular generative model in this context is the stochastic block model (SBM) [22]. Inspired by intriguing conjectures arising from the statistical physics community [29], community detection under the stochastic block model has been studied extensively. As a consequence, the precise information theoretic limits for recovering the underlying communities have been derived, and optimal algorithms have been identified in this setting (for a survey of these recent breakthroughs, see [1]). In reality, the practitioner often has access to additional information in the form of node covariates, which complements the graph information.

We conjecture that the reason for the difference in generalisation between adaptive and non adaptive gradient methods stems from the failure of adaptive methods to account for the greater levels of noise associated with flatter directions in their estimates of local curvature. This conjecture motivated by results in random matrix theory has implications for optimisation in both simple convex settings and deep neural networks. We demonstrate that typical schedules used for adaptive methods (with low numerical stability or damping constants) serve to bias relative movement towards flat directions relative to sharp directions, effectively amplifying the noise-to-signal ratio and harming generalisation. We show that the numerical stability/damping constant used in these methods can be decomposed into a learning rate reduction and linear shrinkage of the estimated curvature matrix. We then demonstrate significant generalisation improvements by increasing the shrinkage coefficient, closing the generalisation gap entirely in our neural network experiments. Finally, we show that other popular modifications to adaptive methods, such as decoupled weight decay and partial adaptivity can be shown to calibrate parameter updates to make better use of sharper, more reliable directions.

In this note, conditions for the existence and uniqueness of the maximum likelihood estimate for multidimensional predictor, binary response models are introduced from a sensitivity testing point of view. The well known condition of Silvapulle is translated to be an empirical likelihood maximization which, with existing R code, mechanizes the process of assessing overlap status. The translation shifts the meaning of overlap, defined by geometrical properties of the two-predictor groups, from the intersection of their convex cones is non-empty to the more understandable requirement that the convex hull of their differences contains zero. The code is applied to reveal the character of overlap by examining minimal overlapping structures and cataloging them in dimensions fewer than four. Rules to generate minimal higher dimensional structures which account for overlap are provided. Supplementary materials are available online.

We propose a framework for deep ordinal regression, based on unimodal output distribution and optimal transport loss. Despite being seemingly appropriate, in many recent works the unimodality requirement is either absent, or implemented using soft targets, which do not guarantee unimodal outputs at inference. In addition, we argue that the standard maximum likelihood objective is not suitable for ordinal regression problems, and that optimal transport is better suited for this task, as it naturally captures the order of the classes. Inspired by the well-known Proportional Odds model, we propose to modify its design by using an architectural mechanism which guarantees that the model output distribution will be unimodal. We empirically analyze the different components of our propose approach and demonstrate their contribution to the performance of the model. Experimental results on three real-world datasets demonstrate that our proposed approach performs on par with several recently proposed deep learning approaches for deep ordinal regression with unimodal output probabilities, while having guarantee on the output unimodality. In addition, we demonstrate that the level of prediction uncertainty of the model correlates with its accuracy.

Spherical data is distributed on the sphere. The data appears in various fields such as meteorology, biology, and natural language processing. However, a method for analysis of spherical data does not develop enough yet. One of the important issues is an estimation of the number of clusters in spherical data. To address the issue, I propose a new method called the Spherical X-means (SX-means) that can estimate the number of clusters on d-dimensional sphere. The SX-means is the model-based method assuming that the data is generated from a mixture of von Mises-Fisher distributions. The present paper explains the proposed method and shows its performance of estimation of the number of clusters.

With growing awareness of societal impact of artificial intelligence, fairness has become an important aspect of machine learning algorithms. The issue is that human biases towards certain groups of population, defined by sensitive features like race and gender, are introduced to the training data through data collection and labeling. Two important directions of fairness ensuring research have focused on (i) instance weighting in order to decrease the impact of more biased instances and (ii) adversarial training in order to construct data representations informative of the target variable, but uninformative of the sensitive attributes. In this paper we propose a Fair Adversarial Instance Re-weighting (FAIR) method, which uses adversarial training to learn instance weighting function that ensures fair predictions. Merging the two paradigms, it inherits desirable properties from both -- interpretability of reweighting and end-to-end trainability of adversarial training. We propose four different variants of the method and, among other things, demonstrate how the method can be cast in a fully probabilistic framework. Additionally, theoretical analysis of FAIR models' properties have been studied extensively. We compare FAIR models to 7 other related and state-of-the-art models and demonstrate that FAIR is able to achieve a better trade-off between accuracy and unfairness. To the best of our knowledge, this is the first model that merges reweighting and adversarial approaches by means of a weighting function that can provide interpretable information about fairness of individual instances.

This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting. We first present the Minimum Mutual Information property, namely the joint measures of two Gaussian measures on Hilbert space with the smallest mutual information are joint Gaussian measures. This is the infinite-dimensional generalization of the Maximum Entropy property of Gaussian densities on Euclidean space. We then give closed form formulas for the optimal entropic transport plan, 2-Wasserstein distance, and Sinkhorn divergence between two Gaussian measures on a Hilbert space, along with the fixed point equations for the barycenter of a set of Gaussian measures. Our formulations fully exploit the regularization aspect of the entropic formulation and are valid both in singular and nonsingular settings. In the infinite-dimensional setting, both the entropic 2-Wasserstein distance and Sinkhorn divergence are Fr\'echet differentiable, in contrast to the exact 2-Wasserstein distance, which is not differentiable. Our Sinkhorn barycenter equation is new and always has a unique solution. In contrast, the finite-dimensional barycenter equation for the entropic 2-Wasserstein distance fails to generalize to the Hilbert space setting. In the setting of reproducing kernel Hilbert spaces (RKHS), our distance formulas are given explicitly in terms of the corresponding kernel Gram matrices, providing an interpolation between the kernel Maximum Mean Discrepancy (MMD) and the kernel 2-Wasserstein distance.

Noisy labeled data is more a norm than a rarity for crowd sourced contents. It is effective to distill noise and infer correct labels through aggregation results from crowd workers. To ensure the time relevance and overcome slow responses of workers, online label aggregation is increasingly requested, calling for solutions that can incrementally infer true label distribution via subsets of data items. In this paper, we propose a novel online label aggregation framework, BiLA, which employs variational Bayesian inference method and designs a novel stochastic optimization scheme for incremental training. BiLA is flexible to accommodate any generating distribution of labels by the exact computation of its posterior distribution. We also derive the convergence bound of the proposed optimizer. We compare BiLA with the state of the art based on minimax entropy, neural networks and expectation maximization algorithms, on synthetic and real-world data sets. Our evaluation results on various online scenarios show that BiLA can effectively infer the true labels, with an error rate reduction of at least 10 to 1.5 percent points for synthetic and real-world datasets, respectively.